Revise Electric field of a point charge for Cambridge 9702 (18.4)

An electric field is a region in space around a charged object where another charged particle would experience a non-contact electrostatic force. We often use the concept of a 'point charge' – an idealised, infinitesimally small charge – to simplify calculations and understand fundamental principles. These fields are radial, meaning they spread outwards from positive charges and converge inwards towards negative charges.

Before we look at field strength, let's remember Coulomb's Law, which describes the electrostatic force between two point charges. This law is crucial because the electric field itself is defined by the force it exerts. The force is proportional to the product of the charges and inversely proportional to the square of their separation.

$F = \frac{1}{4\pi\epsilon_0} \frac{Q_1Q_2}{r^2}$

The electric field strength (E) at a point is defined as the force experienced per unit positive test charge placed at that point. It's a vector quantity, with its direction being the same as the force on a positive test charge. For a single point charge, E decreases rapidly with distance.

$E = \frac{F}{q} \quad \text{or} \quad E = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2}$

While electric field strength describes the force, electric potential (V) describes the energy 'landscape'. It's defined as the work done per unit positive test charge to bring it from infinity (where V=0) to a specific point within the field. Potential is a scalar quantity, meaning it has magnitude but no direction, and its sign matches the source charge.

$V = \frac{W}{q} \quad \text{or} \quad V = \frac{1}{4\pi\epsilon_0} \frac{Q}{r}$

When more than one charge is present, the total electric field and total electric potential at a point are found by summing the contributions from each charge individually. This is known as the Principle of Superposition.

Just as contour lines on a map show points of equal height, equipotential lines (or surfaces in 3D) show points of equal electric potential. For a point charge, these are concentric circles. A key property is that no work is done by the electric field when a charge moves along an equipotential line.

There's a fundamental relationship between electric field strength and electric potential. The electric field strength is essentially the rate at which the potential changes with distance – known as the potential gradient. The field points in the direction of decreasing potential, like a ball rolling down a hill.

$E = |\frac{\Delta V}{\Delta r}|$